Kevin O’Connor
Computation and Consistent Estimation of Stationary Optimal Transport Plans

 In this dissertation, we study optimal transport (OT) for stationary stochastic processes, a field that we refer to as stationary optimal transport. Through example and theory, we argue that when applying OT to stationary processes, one should incorporate the stationarity into the problem directly, constraining the set of allowed transport plans to those that are stationary themselves. In this way, we only consider transport plans that respect the dependence structure of the marginal processes. We study this constrained OT problem from statistical and computational perspectives, with an eye toward applications in machine learning and data science. In particular, we

1. develop algorithms for computing stationary OT plans of Markov chains.
2. extend these tools for Markov OT to the alignment and comparison of weighted graphs.
3. propose estimates of stationary OT plans based on finite sequences of observations.

We build upon existing techniques in OT as well as draw from a variety of fields including Markov decision processes, graph theory, and ergodic theory. In doing this, we uncover new perspectives on OT and pave the way for additional applications and approaches in future work.

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